How can one deal with doubt concerning the purity of the bathroom and bathroom rugs, and the consequent transfer of impurity to household rugs, especially given the presence of severe obsessive doubts in this regard?

The treatment for whisperings (waswas) is to disregard them and not pay them any attention. The default state of all things is purity, including the bathroom floor, unless one is certain that it has been afflicted with impurity.